Relation between Fluid mechanics and Thermodynamics

Is there any valid formula which can apply to both thermodynamics and fluid mechanics, as they are both based on the nature of flow of a substance? If yes, please mention the formula and it's derivation?

Is there any valid formula which can apply to both thermodynamics and fluid mechanics, as they are both based on the nature of flow of a substance? If yes, please mention the formula and it's derivation?

I’m not sure what you’re looking for. The Navier-Stokes equations relate the forces (pressure, and shear) in the fluid to the acceleration of the fluid. Knowing the acceleration you can get the flows. The lost energy generates heat and entropy so you should be able to apply the thermodynamic equations once you know the flows.

And while it may look different than what you may have learned in Thermodynamics it is the same, just written in a form which makes it a bit easier to understand in the context of a fluid mechanics course.

Is there any valid formula which can apply to both thermodynamics and fluid mechanics, as they are both based on the nature of flow of a substance? If yes, please mention the formula and it's derivation?

Those three equations can be written in either a differential way or integral (much like h2oski1326) way. One can also institute "jump conditions" across material or nonmaterial bounderies.

I'm not going to write any formulas- it would take too much time, and there's no need. My go-to book for all this is "Interfacial Transport Phenomena" by Slattery. Brenner and Edwards "Macrotransport Processes" is also very good.

I cannot find the differential form of the energy equation. I expect it to look something like [URL [Broken] equation[/url]

I presume equations involving entropy would be redundant but I came across a paper before which used minimum entropy generation as a principle for deriving empirical forms of convection expressions from computation fluid dynamic techniques.